This book, Di¿erential Geometry: Manifolds, Bundles and Characteristic Classes (Book I-A), is the ¿rst in a captivating series of four books presenting a choice of topics, among fundamental and more advanced, in di¿erential geometry (DG), such as manifolds and tensor calculus, di¿erentiable actions and principal bundles, parallel displacement and exponential mappings, holonomy, complex line bundles and characteristic classes. The inclusion of an appendix on a few elements of algebraic topology provides a didactical guide towards the more advanced Algebraic Topology literature. The subsequent three books of the series are:

Di¿erential Geometry: Riemannian Geometry and Isometric Immersions (Book I-B)

Di¿erential Geometry: Foundations of Cauchy-Riemann and Pseudohermitian Geometry (Book I-C)

Di¿erential Geometry: Advanced Topics in Cauchy-Riemann and Pseudohermitian Geometry (Book I-D)

The four books belong to an ampler book project (Di¿erential Geometry, Partial Di¿erential Equations, and Mathematical Physics, by the same authors) and aim to demonstrate how certain portions of DG and the theory of partial di¿erential equations apply to general relativity and (quantum) gravity theory. These books supply some of the ad hoc DG machinery yet do not constitute a comprehensive treatise on DG, but rather Authors' choice based on their scienti¿c (mathematical and physical) interests. These are centered around the theory of immersions - isometric, holomorphic, and Cauchy-Riemann (CR) -and pseudohermitian geometry, as devised by Sidney Martin Webster for the study of nondegenerate CR structures, themselves a DG manifestation of the tangential CR equations.